WASHINGTON—Supporters of an immigration overhaul, looking for allies in the Republican-led House, concluded months ago that a top prospect would be Majority Whip Kevin McCarthy.

His California district is 37% Hispanic. It is dominated by big farms that rely on immigrant labor. Mr. McCarthy has strong ties to Silicon Valley, where companies are eager for more high-tech visas. His perch as No. 3 House Republican gives him a voice in...